Nonanalytic Points Research Articles

Thermodynamic anomalies near the critical point of steam

Ideas about the character of critical anomalies obtained from the lattice gas model are tested on the thermodynamic properties of steam, using methods developed for nonpolar gases. The critical anomalies in steam are shown to be nonclassical and very similar to those of simple gases. Specifically, the exponents obtained by power-law analysis of the coexistence curve, specific heat C v and compressibility K T are β = 0.347 ±0.005, α = 0.1 ±0.05, γ = 1.20 ±0.05, implying δ = 4.45. Since the critical point is thus a point of nonanalyticity in the thermodynamic behavior, the scaled equation of state may be an appropriate means to describe the critical region. Some results of such a scaled analysis of the PVTdata are presented and shown to be consistent with independently measured vapor pressure and specific heat data and with the exponents quoted. The best value obtained for the critical temperature by scaling is 373.9° C ± 0.05°, values between 0.322 and 0.327 g/cm 3 are obtained for the critical density depending on the property analyzed.

Inclusive cross-sections and discontinuities

The effect of physical-region Landau curves on the inclusive differential cross-section is investigated for a typical class of curves. It is concluded that the differential cross-section is a total discontinuity in the missing mass variableM2 of an amplitude evaluated on an unphysical sheet. There is a point of nonanalyticity at each new singularity.

Analyticity and dipole regularization

It is demonstrated that the modified integration prescription given earlier for the elimination of unphysical asymptotic states in case of a theory with dipole regularization is equivalent to the Lee-Wick integration prescription for poles with complex mass if the dipole is considered as a limiting case of two merging poles, or more precisely, of two merging dipoles with complex conjugate mass. In particular, the thresholds of unphysical processes become points of nonanalyticity in the sense that the function left and right of these points cannot be connected by analytic continuation. The smoothness of this function at the nonanalytic points in the physical region essentially determines the fall-off of «acausal» precursors. It appears that with this procedure even ghost poles below the threshold of the continuum do not lead to a violation of unitarity.

A non-analytic S-matrix

A study is made of theories having the unusual analyticity properties recently proposed by Lee and Wick. A prescription is given for setting up a covariant perturbation-theory expansion of the scattering amplitudes, based on Feynman graphs. It is found that the presence of a complex pole in the upper half plane of the physical sheet leads to points of non-analyticity in the physical region, such that the values of the amplitude to either side of the point are not related by analytic continuation. It is shown how this is compatible with unitarity. The nature of the non-analyticity is not fully determined by unitarity. Neither, in the case of the more complicated graphs, is it fully determined by the perturbation-theory prescription, and some extra constraint must be imposed on the theory to remove the ambiguity. It is shown that the prescription of Lee and Wick has an exactly similar ambiguity, but for their prescription different results are obtained in different Lorentz frames. An estimate is made of the extent to which the theory violates causality, and is found to be too small to measure.

Critical Points and Lattice Vibration Spectra

A method is developed for obtaining directly from group-theoretical and topological considerations a consistent set of critical points belonging to a given secular equation. The treatment includes a detailed discussion of nonanalytic behavior near degeneracies. Comparison with many specific calculations shows that the minimal set so obtained is often the set actually present in the case of short-range forces. The contributions of nonanalytic critical points to the frequency distribution are analyzed. Behavior near critical points is made the basis of a scheme for calculating the main features of frequency distributions from the corresponding minimal sets. An approximate frequency distribution is calculated for aluminum. Possible applications to energy bands are discussed.

On analytic solutions of differential equations in the neighborhood of non-analytic singular points

On analytic solutions of differential equations in the neighborhood of non-analytic singular points